Обработка данных / ElemTreat
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(N0 − 2)∑(xk − x )2 |
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(xk , yks). |
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+1%! !&( 6 $ !# +,& / ! $(+( 6+" 1/+ $ / 2 788!5! ( β ($&( !*$ 6 &!: 2$(6 (#& / 2 &" !* $ +!4!&- (5.43) ! 2 788!-
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b − tα,ν SNb ≤ β ≤ b + tα,ν SNb . |
(5.44) |
+1%! !&( 6 $ !# +,& / ! $(+( 6+" '$ ) 6& / 4+ &( γ +!-
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(5.45) |
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∑(xk − x ) |
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k =1 |
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$ !# +,&-0 ! $(+ 6+" '$ ) 6& / 4+ &( γ $- (9( #'" '+ -
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g − tα,ν SNg ≤ γ ≤ g + tα,ν SNg , |
(5.46) |
61
/6 4!'+ '# & 0 '$ ) 6- ν = N0 – 2, ( $ +!4!&( g $-4!'+" #'" $# 0 8 31+ !* (5.39).
, '# !3 6 $ !# +,&-0 ! $(+ 6+" *&(4 &!" 81&25!! Y(x) # !*$ +,& / *&(4 &!" ( /13 ( x. # # ! $(+ 3 9 # )-#,6'#($+ & 6$13" & ($ &'#$(3!, !' +,*1:;!3! # # 9 2 788!5!
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≤ Y (x) |
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(x)+ tα,ν SNX , |
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Y (x)− tα,ν SNX |
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(x − x )2 |
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(5.48) |
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k =1 |
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( 2'!3! 1:; 0 81&2- |
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&(4 &! 81&25!! Y (x) $-4!'+" #'" |
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5!! (5.12) ' !' +,* $(&! 3 8 31+ (5.39). |
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4 $!6& , 4# $ +!4!&( SNX )-'# $ * ('#( # |
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xmax ' #$ #'#$ && 3!&!3(+,& ! 3(2'!3(+,& *&(4 &!" !* 3& 9 - '#$( xk (k = 1, …N). +" ( /13 $ x > xmax ! x < xmin 6 $ !# +,- &-0 ! $(+ (5.47) '#(& $!#'" #(2!3 / 3&-3, 4# +!%( #'" 8!*!4 -
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N0 = Nm , |
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(5.49) |
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N ∑xk2 − ∑xk |
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(5.50) |
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N ∑xk2 − ∑xk |
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(5.51) |
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Q0 = m∑(yk − yk ) |
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!6& , 4# 8 31+- (5.50) #+!4(:#'" # $- (9 &!0 (5.11) § 5.2
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(5.52) |
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/6( '133( 2$(6 (# $ (5.51) 3 9 # )-#, 6'#($+ &( $ 2 (#2 0 8 3 :
Q0 = mQ1 + Q2 . |
(5.53) |
63
(2!3 ) (* 3, 6+" (''3(# !$( 3 / '+14(" mk = m = const $--(9 &!" 6+" 6 $ !# +,&-. ! $(+ $ (5.44), (5.46), (5.47) 3 9& '-
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(Nm − 2)∑(xk − x )2 |
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(5.55) |
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§ 5.6. ( % %
('# (/ (*6 4(; , 4 3 . # + ', )-) # #!4 '2(" 8!*!2( 1#$ - 96( #, 4# *($!'!3 '#, !''+ 61 3-. 8!*!4 '2!. $ +!4!& X ! Y "$+" #- '" & +!& 0& 0.
+ 9 &! 8!*!2(-72' !3 (# ( )+ /4( #'" # 3, 4# 3& /! 81&25! &(+,&- *($!'!3 '#!, $'# 4(:;! '" $ 8!*!2 , + /2 !$ - 6"#'" 2 +!& 0& 0 !' +,* $(&! 3 (+/ ) (!4 '2!. ) (* $(&!0. - '+ #(2 0 «+!& ( !*(5!!» 81&25!! 3 9& !' +,* $(#, 3 # 6 &(!- 3 &,%!. 2$(6 (# $ $ $-% !*+ 9 && 0 8 3 . +14 &&- 5 &2! 2 - 788!5! $ +!& 0& 0 *($!'!3 '#! *(# 3 '4!#-$(:#'" $ ( (3 #-- !'. 6& 0 81&25!! ) (#&-3 (+/ ) (!4 '2!3 ) (* $(&! 3.
64
(''3 # !3 $ 2(4 '#$ #! !4& / !3 ( 72' & &5!(+,&1: *($!- '!3 '#,, 2 # (" 4('# !'-$( # (*& ) (*&- 8!*!4 '2! 5 ''-.4('#& '#!, &(+!4! 6!''! (#!$&-. '!+ $ '!'# 3 96( # 72' - & &5!(+,& 1)-$(&! 3 .(&!4 '2 0 7& /!! ! & 2 # -. 6 1/!. 8!- *!4 '2!. $ +!4!& ' # 4 &! 3 $ 3 &!.
1'#, 6$ 8!*!4 '2! $ +!4!&- X ! Y '$"*(&- *($!'!3 '#,: $!6(:
Y = A0 exp(−B0X). |
(5.57) |
1&25!" ' 6 9!# 6$( ( (3 # ( A0 ! B0, 2 # - '# "&&- ! - + 9!# +,&-, & !. 4!'+ &&- *&(4 &!" & !*$ '#&- 72' !3 (# 1.
. 6 72' !3 ( $ 6"#'" !*3 &!" $ +!4!&- Y 6+" (*&-.
*&(4 &!0 $ +!4!&- X ! +14(:#'" N ( 4!' + xk, yk (k = 1,...,N).+" 2 (#2 '#! !*+ 9 &!" $ 7# 3 ( (/ (8 + 9!3, 4# $' *&(4 &!"
( /13 ( xk (*+!4&-.
16 3 !'2(#, 4!'+ $- *&(4 &!" ( (3 # $ A0 ! B0, 2 # - ' #- $ #'#$1:# &(!+14% 31 !)+!9 &!: 81&25!! (5.57) 2 -#&-3 -
*1+,#(#(3, !' +,*1" $-% !*+ 9 &&-0 3 # 6 &(!3 &,%!. 2$(6 (# $.$ 6 3 & $1: 3 &&1: Z, 2 # 1: 6 +!3 2(2 &(#1 (+,&-0
+ /( !83 $ +!4!&- Y:
Z = ln(Y) . |
(5.58) |
) (* $(&! (5.58) $ (;( # 72' & &5!(+,&1: *($!'!3 '#, (5.57) $ +!& 0&1: 81&25!::
Z = β* X + γ* . |
(5.59) |
'# "&&- $ +!4!&- β* ! γ* 6& *&(4& $- (9(:#'" 4 * ( (- 3 # - !'. 6& 0 81&25!! (5.57)
β* = −B0 , γ* = ln(A0) . |
(5.60) |
+" 2(96 / !*3 && / *&(4 &!" yk $-4!'+!3 / &(#1 (+,&-0
+ /( !83 zk = ln(yk), $ ' #$ #'#$!! ' $-) (&&-3 ) (* $(&! 3 (5.58). (5!" "$+" #'" 6 1'#!3 0, # 31 4# $' *&(4 &!" 81&2-
5!! (5.57) + 9!# +,&-. *1+,#(# 3- +14!3 N ( '+ 61:;!. 4!' +:
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xk , zk (k = 1, ..., N) . |
(5.61) |
, 3 9& 6+" ( *&(4 &!0 (5.61) ! ( |
2'!3! 1:; 0 81&2- |
5!! (5.59) !3 &!#, 3 # 6 &(!3 &,%!. 2$(6 (# $ ' /+('& 3 # 6!2 ,!'(&& 0 $ 6-61;!. ( (/ (8(.. (''3(# !$( 3 3 '+14( &(!-
+14%!3 !)+!9 &! 3 ( (3 # $ β* ! γ* 81&25!! (5.59) "$+":#'" $ +!4!&- b* ! g*, $-4!'+" 3- 8 31+(3 (5.11), $ ($-. 4('#".
2 # -. $' *&(4 &!" yk *(3 & &- &( ' #$ #'#$1:;! &(#1 (+,&-
+ /( !83- zk = ln(yk). (# 3 $- (*!3 !'2 3- ( (3 # - A0 ! B0 4 - |
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(5.58). ! 7# 3 3- |
+14!3 !)+!9 &&- *&(4 &!" |
( (3 # $ A0 ! |
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B0 72' & &5!(+,& 0 81&25!! (5.57): |
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= exp(g |
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(5.62) |
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B0 = −b |
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($ &'#$( A0 ≈ A0 |
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B0 ≈ B0 "$+":#'" |
!)+!9 &&-3!, #(2 2(2 $ - |
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+!4!&- b* ! g* (''4!#(&- *1+,#(#(3 !*3 &!0 xk, yk, 2 # - ' 6 9(# '+14(0&- / %& '#!.
+ 6 $(# +,& , 2 3 !)+!9 &&-. |
*&(4 &!0 (5.62) |
( (3 # $ |
81&25!! (5.57), & ). 6!3 $-4!'+!#, |
/ %& '#! !. 6 + &!". |
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+" ( (3 # ( B0 )+ 3( %( #'" '# . /+('& $# 31 (- $ &'#$1 !* (5.62), 7# # ( (3 # ' $ (6( # ()' +:#& 0 $ +!4!& ' 1/+ $-3 2 788!5! 3 +!& 0& 0 81&25!! (5.59). # *&(4!#, 4# -
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2+ & &! Sb 6 +" #'" $- (9 &! 3 (5.16). '# '#$ && , 4# 6$(-!# +,& 3- *(6( 3 6 + &&1: 6 $ !# +,&1: $ "#& '#, α ! !*- $+ 2( 3 !* #()+!5 ' #$ #'#$1:; *&(4 &! 2 788!5! ( #,:6 (.!'+ '# & 0 '$ ) 6- $ 7# 3 '+14( ) #'" ($&-3 N – 2. &(4 / $ ", 6 $ !# +,&-0 ! $(+ 6+" ( (3 # ( B0 '# !#'" !6 !4&
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66
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= exp(g*) "$+" #'" & +!& 0&-3, 4# *(# 16&" # $-4!'- |
(* $(&! A |
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+ &! ' 6& 2$(6 (#!4& / #2+ & &!" '+14(0& 0 $ +!4!&- A0 .
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$-4!'+!3 / (&!5- 6 $ !# +,& / ! $(+( 6+" ( (3 # ( γ*, ' /+('- & ); 31 $- (9 &!: (5.20) 6+" '$ ) 6& / 4+ &( +!& 0& 0 81&25!!:
g* − tα, N–2 Sg ≤ γ* ≤ g* + tα, N–2 Sg . |
(5.63) |
, 7# 6$ 0& & ($ &'#$ 6$ /& 3 72' & &5!(+,& 31
) (* $(&!:. +14 &&- *&(4 &!"
exp( g* − tα, N-2 Sg ) ; exp( g* + tα, N-2 Sg ) |
(5.64) |
*(6(:# / (&!5- & 2 # / ! $(+(. '+! ! $(+ (5.63) &(2 -$( # ( (3 # γ* ' $ "#& '#,: α, # ! $(+ (5.64) &(2 -$( # ( (3 # A0 = exp(γ*) ' # 0 9 $ "#& '#,:. 7# 31 ! $(+ (5.64) 3 9& !' +,* $(#, 2(2 6 $ !# +,&-0 6+" !'2 3 / ( (3 # ( A0 ! *(6(&- & 0 6 $ !# +,& 0 $ "#& '#! α. (2!3 ) (* 3, 3- $-4!'+!+! / (- &!5- 6 $ !# +,& / ! $(+(, & !' +,*1" ' 6& 2$(6 (#!4& / #-
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68
§ 5.7. & ! # ! * (
+" !++:'# (5!! !3 & &!" 3 # 6!2!, !*+ 9 && 0 $ 6-61;!. ( (/ (8(. 6(&& 0 /+($-, !$ 6 3 & '2 +,2 (2#!4 '2!. !3 $.
1. +() (# & 0 () # $ / 21 '( « 6 + &! 3 - 3 ( !& 5!! # + #& '!# +,& & 5 (+,&-. ' 0» !' +,*1 #'" 1'#(& $2(, $ 2 &'# 125!: 2 # 0 $. 6!# 6$ % &&(" &( $ + 2
+(#8 3( ' # 3" 5 ! 1:;!3! / 1*(3!. + , 3 3 !& 5!! J 2 # / 6'# !# !*3 !#,, 3 ;( #'" &( +(#8 31. $ # +(#8 3- &( & 2 # -0 1/ + 96( # 2 1#!+,&- 2 + )(&!". *3 -
&! ! 6( 2 1#!+,&-. 2 + )(&!0 |
*$ +" # $-4!'+!#, 3 3 |
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#!4 '2!0 (&(+!* 2 + )(# +,& / |
5 ''( |
2(*-$( #, 4# 3 - |
3 !& 5!! # +( J '$"*(& ' ! 6 3 2 + )(&!0 |
+(#8 3- T '+ - |
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61:;!3 ) (* 3: |
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J = ' T 2 − J0 , |
(5.65) |
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/6 J0 — 3 3 !& 5!! +(#8 3- ' # 3" / 1*(3! ! ' — & 2 # -0'# "&&-0 2 788!5! .
+ 6 $(# +,& , &( &(4(+,& 3 7#( () #- '#16 (3 & ). 6!3
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+" 6 + &!" 7#!. & !*$ '#&-. $ 6"#'" !*3 &!" ! 6( 2 1#!+,&-. 2 + )(&!0 ' !' +,* $(&! 3 $ 2(4 '#$ # + 6!'2 $, 2 # - 2 (2'!(+,& 3 ;(:#'" &( +(#8 31. 3 !& 5!! 6& 6& / 6!'2( #& '!# +,& ' )'#$ && 0 '! '!33 # !! 6 +" #'" / 3('- ' 0 ! (6!1' 3. *3 &! ! 6( 2 1#!+,&-. 2 + )(&!0 )-+ !'(- & $ !3 6+" # !8!+" & / 6$ '( $ /+($ 3.
1'#, $ 5 '' !*3 &!0 )-+! +14 &- *1+,#(#-, !$ 6 &- &- $ $-. 6$1. '# 2(. #()+!5- 1.
69
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! 1 |
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!'+ -# $ N = 4 |
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3 |
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1 |
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133- |
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-#( (k) |
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k (') |
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1,41 |
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1,525 |
1,63 |
1,725 |
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Jk (2/ 3 |
2 |
) |
4,75 10 |
−3 |
9,9 10 |
−3 |
15 10 |
−3 |
20 10 |
−3 |
4,965 10− |
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2 |
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k2 (c2) |
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1,988 |
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2,326 |
2,657 |
2,976 |
9,946 |
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k4 (c4) |
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3,953 |
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5,409 |
7,059 |
8,854 |
25,275 |
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Jk k2 |
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9,443 10− |
2,302 10− |
3,985 10− |
5,951 10− |
0,1318 |
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(2/ 32 c2) |
3 |
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2 |
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/+('& (5.65), *($!'!3 '#, 3 961 2$(6 (# 3 ! 6( 2 + )(&!0 +(#8 3- T2 ! 3 3 3 !& 5!! # +( J "$+" #'" +!& 0& 0 ! 8 -
3(+,& ' $ (6( # ' 81&25! 0 $!6( (5.7). ! 7# 3 2 788!5! ' !3 -# '3-'+ 1/+ $ / 2 788!5! ( β, ( 3 3 !& 5!! +(#8 3- J0 — '3-'+ '$ ) 6& / 4+ &( γ +!& 0& 0 81&25!! (5.7), $*"# / ' ) (#&-3
*&(2 3. '# '#$ && , 6+" &(. 96 &!" & !*$ '#&-. '# "&&-. ' ! J0 '+ 61 # !3 &!#, $-% !*+ 9 &&-0 3 # 6 &(!3 &,%!. 2$(6 (# $.
$(6 (#!4&(" 81&25!" (5.65) $ (;( #'" $ +!& 0&1:, '+! ( /1- 3 3 +(/(#, 2$(6 (# ! 6( 2 + )(&!0. 7# 31, 4# )- !3 - &!#, 3 # 6 ('4 #(, !*+ 9 &&-0 $ § 5.2 ! § 5.3, '+ 61 # $ 2(4 '#$ $ +!-
4!& xk !' +,* $(#, 2$(6 (#- !*3 &&-. *&(4 &!0 ! 6(. -4!'-
+ &&- *&(4 &!" xk = k2 (k =1, …4) *( !'(&- $ # #, 0 '# 2 #()+. 1.
'#(+,&-. '# 2(. #()+!5- 12(*(&- !*$ 6 &!", & ). 6!3- 6+" ('4 #( 4 #- . '133, $. 6";!. $ 8 31+- (5.11). '+ 6&!0 '# +) 5 #()+!5- ' 6 9!# 7#! 4 #- '133-, !$ 6 &&- *6 ', 6+" 2 +" *1+,#(# $ 3 91# 4&-. $-4!'+ &!0.
6(&&-3 #()+!5- 1 $ 6"#'" $-4!'+ &!" $ +!4!& b ! g ' - 3 ;,: 8 31+ (5.11):
b = 0,0154 , g = − 0,026 .
# *&(4!#, 4# !'2 3- ( (3 # - !3 :# '+ 61:;! *&(4 &!"
' ≈ 0,0154 (2/ 32/c2) , J0 ≈ 0,026 (2/ 32). |
(5.66) |
70